The visitors were in the game throughout, and went close to finding the net on a number of occasions. However, ultimately the champions of England prevailed, and made it through to the next stage of the competition.
Manager Mark Robins made one change from the 1-1 draw with Wigan Athletic, with Murray Wallace coming in for Niall Canavan, to make his first start since joining permanently from Huddersfield.
The rest of the Iron lineup was unchanged. Luke Daniels started in goal, and Jordan Clarke, Wallace, David Mirfin and Scott Laird played in a back four in front of him. Neal Bishop, Stephen Dawson and Jamie Ness were the midfield three, and Paddy Madden and Kevin van Veen were provided support by Luke Williams.
After making a solid start to the match, United fell behind on 13 minutes. Chelsea captain Branislav Ivanovic had time to deliver from the right, and when he did, he picked out Blues frontman Diego Costa, who finished well with a left footed volley.
And shortly after getting their noses in front, the hosts quickly went looking for a second. However, they were denied by a superb piece of goalkeeping by Daniels. The Iron stopper went full stretch to tip a Cesc Fabregas strike over the crossbar.
He had to show good reactions once more, 20 minutes in. This time he diverted a Pedro effort out from a corner, after the Spaniard had struck from a tight angle.
Robins' side were keen to respond, and mid-way through the half they went close to doing so. After Ness had whipped in a dangerous corner from the right, the ball fell to Dawson on the edge of the penalty area, and the Irishman hit an shot which looked to be heading for the top corner of the goal. But he was to be frustrated, as the Chelsea defence stood firm and blocked his effort.
As the half wore on, the Iron gradually grew more and more into the match. And 30 minutes in, Williams had United's first effort on target. The attacker initially did well to pinch the ball from Costa, and then played a neat one-two with Laird before forcing Begovic into a smart save.
Williams was causing real problems, and minutes before the break he had United's best opportunity. The Iron number seven beat two Chelsea defenders, but a heavy touch in the box prevented him from testing Begovic once again, as Kurt Zouma got across to avert the danger.
Craig Pawson's half-time whistle would have been something which would have come as something of a frustration for the Iron, as they were threatening to peg Chelsea back before the interval.
The visitors picked up where they left off after the restart, with van Veen driving at the heart of the Blues' back four. And the Dutchman might have felt aggrieved not to be awarded a penalty, as he looked to be tripped by Ramires as he entered the box.
But despite being camped in the hosts' half for the majority of the second period, the Iron were hit with a sucker punch 22 minutes from time. Substitute Ruben Loftus-Cheek stole a yard at the near post, and drove the ball under Daniels, after a low cross from the left.
The goal did little to hurt United's confidence, as they instantly went looking to get back into the game. Van Veen, who'd been a constant threat throughout, looked to have halved the deficit with a free-kick from the right, but he could only find the side-netting, much to the frustration of the Dutchman.
Even with little left on the clock, the Iron kept pressing. Full-back Clarke was the next to try his luck with a thunderous strike from the edge of the area, which Begovic tipped onto the post, and out for a corner kick.
In spite of United's best efforts, they couldn't find a consolation goal. However, it was a performance which the Iron players could be proud of, as they tested the hosts right up until the final whistle.
Chelsea: Begovic, Ivanovic, Zouma, Cahill, Azpilicueta, Ramires, Fabregas, Willian, Oscar (Loftus-Cheek 46), Pedro (Kenedy 71), Diego Costa (Traore 86).
Substitutes not used: Courtois, Terry, Baba Rahman, Mikel.
Scunthorpe United: Daniels, Clarke, Mirfin, Wallace, Laird, Dawson (King 79), Ness, Bishop, Williams (Adelakun 79), Madden, van Veen (Wootton 81).
Substitutes not used: Anyon, Wiseman, Canavan, McAllister